Secondary seed dispersal in the Klausmeier model of vegetation for sloped semi-arid environments

The effects of secondary seed dispersal on the dynamics of banded vegetation are investigated in the framework of a generalized version of one of the easiest tools for the description of pattern formation along the hillsides of semi-arid environments: the Klausmeier model. The generalization here considered consists in augmenting the evolution equation for the vegetation biomass with an advection term mimicking the anisotropic dispersal of seeds by overland flow. Linear stability analysis is used to deduce the threshold condition for the occurrence of wave instability as well as to obtain approximated explicit expressions for some key quantities: pattern speed, locus of stationary patterns and excited wavenumber. The generalized model is also integrated numerically considering two different sets of initial conditions that yield pattern dynamics originating from degradation of uniform vegetation or colonization of bare ground. These numerical simulations are performed to corroborate the analytical predictions, to characterize more deeply the propagating character of the edges of vegetation patches and to emphasize how distinct initial conditions may lead to significantly different ecological scenarios.